Seminars at City, University of London

24.01.2012 (Tuesday)

A Ricci soliton is a generalisation of an Einstein metric which
evolves in a very simple way under the Ricci flow. We discuss ways of
producing examples of Ricci solitons by looking for solutions with a high
degree of symmetry

25.10.2011 (Tuesday)

This talk will give a pedagogical account of the role played
by integrability and instantons in N=2 SUSY gauge theories leading
to recent understanding of what it means to quantize the integrable
system.

19.10.2011 (Wednesday)

Dimer models are typically studied in condesed matter
physics and combinatorics. The correspondence between
dimer models, toric Calabi-Yaus and quiver gauge theories
on D-branes has had a profound impact in areas ranging
from string phenomenology to mathematics. Today I will discuss
a recently discovered correspondence between dimer models
and integrable systems.

19.10.2011 (Wednesday)

I will discuss scattering amplitudes in N=2,4,8 SYM in three-dimensions, concentrating on the N=8 case, with an emphasis on which properties of the
N=4, D=4 SYM amplitudes survive under dimensional reduction. The on-shell
supersymmetry algebra makes the SO(N) symmetry of the amplitudes manifest,
while the Lagrangian displays only manifest SO(N-1) symmetry. I will also discuss
the possibility of non-local Yangian-type symmetry, connections to BLG,
and some perspectives on loop level results. Based on 1103.0786 / 1109.2792.

18.10.2011 (Tuesday)

Abstract :
Feynman Graph counting in Quantum Field Theory (QFT) can be formulated in terms
of symmetric groups. This leads to expressions for
graph counting and symmetry factors in terms of topological transition
amplitudes for strings with a cylinder target, related to two dimensional
topological field theory. The details of the interactions in the QFT
are encoded in the boundary conditions which specify
how the strings wind around circles. The QFTs discussed include scalar field theories and QED, where there is no large gauge group.

11.10.2011 (Tuesday)

We shall talk generally about the state of the art in set theory and try to explain to what extent the independence results in set theory influence our understanding of mathematical foundations. The talk will start rather generally and will build up to describe some current research directions.

04.10.2011 (Tuesday)

The importance of adequately modeling credit risk has once again been
highlighted in the recent financial crisis. Defaults tend to cluster
around times of economic stress due to poor macro-economic conditions,
but also by directly triggering each other through contagion. Although
credit default swaps have radically altered the dynamics of contagion
for more than a decade, models quantifying their impact on systemic risk
are still missing. Here, we examine contagion through credit default
swaps in a stylized economic network of corporates and financial
institutions. We analyse such a system using a stochastic setting, which
allows us to exploit limit theorems to exactly solve the contagion
dynamics for the entire system. Our analysis shows that CDS, when used
to expand banks' loan books (arguing that CDS would offload the
additional risks from banks' balance sheets), can actually lead to
greater instability of the entire network in times of economic stress,
by creating additional contagion channels. This can lead to considerably
enhanced probabilities for the occurrence of very large losses and very
high default rates in the system. Our approach adds a new dimension to
research on credit contagion, and could feed into a rational
underpinning of an improved regulatory framework for credit derivatives.

08.03.2011 (Tuesday)

In algebraic geometry and string theory there has been a lot of recent work on so-called wall-crossing phenomena for Donaldson-Thomas invariants.
In this talk we will study a baby example of wall-crossing, which already has some non-trivial consequences. I will not assume any previous knowledge of algebraic geometry, just some basic properties of the category of modules over a ring.

25.01.2011 (Tuesday)

The equations for Abelian Higgs vortices (magnetic flux vortices) on a plane
or a more general surface are generally not integrable, but for vortices
on a hyperbolic plane of curvature -1/2 they are. This talk will
present (almost explicit) vortex solutions on certain compact hyperbolic
surfaces. Also to be discussed are two asymptotically solvable problems
for vortices: the effective vortex motion on a large surface with
small curvature, and the structure of vortex solutions on a small
surface where the vortices are about to dissolve (and the equations
linearize).
These results (obtained with N. Rink and with N. Romao) bring vortex
theory closer to classical results on the complex and metric geometry
of Riemann surfaces.

07.12.2010 (Tuesday)

Let us say that two conjugacy classes of a group commute if they contain representatives that commute. When G is a finite group with a normal subgroup N such that G/N is cyclic, one can use this definition, together with Hall's Marriage Theorem, to describe the distribution of the conjugacy classes of G across the cosets of N. I will give an overview of this result, and then talk about some more recent work on commuting conjugacy classes in symmetric and general linear groups. This talk is on joint work with John Britnell.

23.11.2010 (Tuesday)

According to AdS/CFT a remarkable correspondence exists between strings in AdS5 x S5 and operators in N=4 SYM. A particularly important case is that of fast-spinning folded closed strings and the so called twist-operators in the gauge theory. This is a remarkable tool for uncovering and checking the detailed structure of the AdS/CFT correspondence and its integrability properties.
In this talk I will show how to match the expression of the anomalous dimension of twist operators as computed from the quantum superstring with the result obtained from the Bethe ansatz of SYM. This agreement resolves a long-standing disagreement between gauge and string sides of the AdS/CFT duality and provides a highly nontrivial strong coupling test of SYM integrability.

16.11.2010 (Tuesday)

Some aspects of the modular representation theory of a finite group can be
described by a tree. Such trees have been determined for almost all finite
simple groups, but some cases remain unknown. Starting from the example of
the group SL2(q) I will explain how geometric methods can be used to solve
this problem for finite reductive groups.

09.11.2010 (Tuesday)

I will speak about a new research project called 'Evolution as an information dynamic system', which involves collaboration between four universities in the United Kingdom. This is a three year project started this year, 2010, and its aim is to develop new understanding of information dynamics in evolution and biology. In particular, we are going to derive new optimality conditions for some evolutionary operators, such as mutation and recombination. Evolutionary states will be represented by probability measures on the space of genetic sequences, and different operators produce different evolution of the states. We define the optimality conditions for evolution based on the maximisation of utility (or fitness) of information principle. The optimal evolution in this sense achieves the shortest 'information distance', and it can be different from an evolution optimal in another sense, such as the shortest convergence time. We argue that the former achieves a better adaptation of organisms living in a dynam
ic environment. I will present several early results related to the optimisation of mutation rate parameter. I will review these results in the light of the classical theories of adaptation (e.g. Fisher's geometric model) and error threshold. Then I will outline some future theoretical and experimental work of the project.

02.11.2010 (Tuesday)

Many if not all models of disease transmission on networks can be
linked to the exact state-based Markovian formulation. However the large
number of equations for any system of realistic size limits their
applicability to small populations. As a result, most modelling work relies
on simulation and pairwise models. In this talk, for a simple
SIS dynamics on an arbitrary network, we formalise the link between a well
known pairwise model and the exact Markovian formulation and we formalise
lumping and its direct link to graph automorphism. Lumping is a powerful
technique that exploits graph symmetry and allows to keep the model exact
while considerably reducing the number of equations. Finally, for pairwise
model two different closures are presented, one well established and one
that has been recently proposed. The closed dynamical systems are solved
numerically and the results are compared to output from individual-based
stochastic simulations. This is done for a range of networks
with the same average degree and clustering coefficient but generated using
different algorithms. It is shown that the ability of the pairwise system
to accurately model an epidemic is fundamentally dependent on the underlying
large-scale network structure. We show that the existing pairwise models
work well for certain types of network but have to be used with caution as
higher-order network structures may compromise their effectiveness.
Keywords: network, epidemic, Markov chain, moment closure.

19.10.2010 (Tuesday)

Somos sequences are generated by a rational recurrence, which is specified by a quadratic relation
between adjacent iterates. Michael Somos noticed that, for some special choices of initial values, such recurrences
could unexpectedly produce sequences of integers. Examples of Somos sequences were known somewhat earlier in
number theory, from Morgan Ward's elliptic analogues of Fibonacci and Lucas sequences. In algebraic combinatorics,
Somos recurrences provide a basic example of the Laurent phenomenon, which is a cornerstone of Fomin and Zelevinsky's
theory of cluster algebras. This introductory talk reviews the history of Somos sequences and their connections
with these and other areas of mathematics and theoretical physics, including solvable statistical mechanics (the hard hexagon
model) and discrete integrable systems (QRT maps and the discrete Hirota equation).

13.10.2010 (Wednesday)

We study the Renyi entropy of the one-dimensional XYZ spin-1/2 chain in the entirety of its phase diagram. The model has several quantum critical lines corresponding to rotated XXZ chains in their paramagnetic phase, and four tri-critical points where these phases join. Two of these points are described by a conformal field theory and close to them the entropy scales as the logarithm of its mass gap. The other two points are not conformal and the entropy has a peculiar singular behavior in their neighbors, characteristic of an essential singularity. At these non-conformal points the model undergoes a discontinuous transition, with a level crossing in the ground state and a quadratic excitation spectrum. We propose the entropy as an efficient tool to determine the discontinuous or continuous nature of a phase transition also in more complicated models.